Lax representations via twisted extensions of infinite-dimensional Lie algebras: some new results

Abstract

We apply the technique of twisted extensions of infinite-dimensional Lie algebras to find new 3D integrable pdes related to the deformations of Lie algebra R N [s] ⊗ w with N = 1, 2 as well as to the Lie algebra h ⊕ w, where R N [s] is the algebra of truncated polynomials of degree N , w is the Lie algebra of polynomial vector fields on R and h is the Lie algebra of polynomial Hamiltonian vector fields on R 2 .

Integrable partial differential equations and Lie--Rinehart algebras

“…Therefore the problem of finding intrinsic properties that ensure existence of a Lax representation for a given pde is of great interest. In the series of papers [20] - [25] we proposed the method to attack this problem via the technique of twisted extensions of Lie algebras of symmetries of the pdes under the study. This approach is of limited scope and can not be used in some examples.…”

“…In the present paper we generalize the approach of [20] - [25] for the Lie-Rinehart algebras. We discuss twisted extensions of Lie-Rinehart algebras as well as extensions by appending an integral of a non-trivial 1-cocycle.…”